Hyperons in Neutron Stars across the observed mass range: Insights from realistic $\Lambda$-N and $\Lambda$-$\Lambda$ interactions within a Microscopic Framework

We investigate the equation of state (EOS) and macroscopic properties of neutron stars (NSs) and hyperonic stars within the framework of the lowest order constrained variational (LOCV) method, extended to include interacting $\Lambda$ hyperons. The nucleon-nucleon interaction is modeled using the AV18 potential supplemented by Urbana three-body forces, while $\Lambda N$ and $\Lambda \Lambda$ interactions are described by realistic spin- and parity-dependent potentials fitted to hypernuclear data. Cold, charge-neutral, and $\beta$-equilibrated matter composed of neutrons, protons, electrons, muons, and $\Lambda$ hyperons is considered. We compute particle fractions, chemical potentials, the EOS, speed of sound, tidal deformability, and stellar structure by solving the Tolman-Oppenheimer-Volkoff equations, and compare our results with recent NICER and gravitational-wave observations. The inclusion of $\Lambda$ hyperons leads to EOS softening, reducing the maximum NS mass from $2.34M_\odot$ to $2.07M_\odot$, while keeping it consistent with the $2M_\odot$ mass constraint. At $1.4M_\odot$, the model satisfies observational limits on radius and tidal deformability, with the $\Lambda$ onset occurring below this mass. Comparison with other microscopic and relativistic mean-field models shows that our EOS remains consistent with the allowed pressure-energy density range, while also permitting even canonical-mass NSs of about $1.4M_{\odot}$ to accommodate hyperons. These results suggest that hyperons can appear in NSs across the observed mass range without violating current astrophysical constraints, and that the extended LOCV method provides a consistent, microscopic approach to modeling dense hypernuclear matter.